Question: Given $ m \angle QPR = 6x - 60$, $ m \angle RPS = 3x - 23$, and $ m \angle QPS = 34$, find $m\angle RPS$. $P$ $Q$ $S$ $R$
Answer: From the diagram, we see that together ${\angle QPR}$ and ${\angle RPS}$ form ${\angle QPS}$ , so $ {m\angle QPR} + {m\angle RPS} = {m\angle QPS}$ Substitute in the expressions that were given for each measure: $ {6x - 60} + {3x - 23} = {34}$ Combine like terms: $ 9x - 83 = 34$ Add $83$ to both sides: $ 9x = 117$ Divide both sides by $9$ to find $x$ $ x = 13$ Substitute $13$ for $x$ in the expression that was given for $m\angle RPS$ $ m\angle RPS = 3({13}) - 23$ Simplify: $ {m\angle RPS = 39 - 23}$ So ${m\angle RPS = 16}$.